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arXiv · 2609.36613

Block-Transitive $5$-$(v,k,2)$ Designs with $k$ divides $v$

Huijiao Hu·Zheng Huang·Shouqiang Shen
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Abstract

The classification of block-transitive 5-designs remains an open problem. The additional parameter condition that $k$ divides $v$ is called the Camina-Gagen condition. In this paper, we investigate block-transitive simple $5$-$(v,k,2)$ designs satisfying the Camina-Gagen condition. Using the classification of finite $2$-homogeneous permutation groups, we consider the affine and almost simple cases separately. We prove that no such design admits a block-transitive automorphism group of affine type. For the almost simple case, up to isomorphism, there are exactly two possibilities: a $5$-$(12,6,2)$ design admitting PGL(2,11) as a block-transitive automorphism group and a $5$-$(24,8,2)$ design admitting PGL(2,23) as a block-transitive automorphism group.

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Huijiao Hu, Zheng Huang, Shouqiang Shen. 2026-09-29. Block-Transitive $5$-$(v,k,2)$ Designs with $k$ divides $v$. https://arxiv.org/abs/2609.36613

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